{
  "abstract": "Using a convex pentagonal monotile belonging to the Type 5 family, we investigate the relationships among the hat tile, turtle tile, and Tile$(1, 1)$. By applying Sugimoto's Perspective and Amfirifma's Perspective, we obtain four types of concave polygons, AH-tile, BH-tile, AT-tile, and BT-tile, each having Heesch number 1 under the conditions considered. We show that these polygons correspond to clusters used to generate the non-periodic tilings $\\mathscr{T}_h$ and $\\mathscr{T}_s$. We further discuss the possibility that tile sets consisting of pairs selected from these polygons may correspond to $\\textit{ASPmr}\\{\\text{A-tile}, \\text{B-tile}\\}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09603",
  "author_total": 1,
  "authors": [
    {
      "name": "Teruhisa Sugimoto"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4wbfg7em4sejq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09603",
  "fulltext_url": null,
  "id": "4wbfg7em4sejq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09603",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2505|Quasicrystal Structures and Properties"
    ]
  ],
  "openalex": "W7212155050",
  "pdf_url": "https://arxiv.org/pdf/2609.09603",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4wbfg7em4sejq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09603",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09603"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Tile sets consisting of two types of concave polygons derived from periodic tilings corresponding to non-periodic tilings with hat and turtle tiles",
  "title_text": "Tile sets consisting of two types of concave polygons derived from periodic tilings corresponding to non-periodic tilings with hat and turtle tiles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/4wbfg7em4sejq",
  "venue": "arXiv",
  "versions": []
}